by Jyotiraditya
November 2018 - January 2019

from Medium Website

Spanish version







Part 1

November 02, 2018




whether waves, particles or anything in between,

have properties that define what they are.

But they require a stage on which to

interact and play out the Universe's story.

Image credit: Wikimedia Commons user Maschen.



When it comes to understanding the Universe, there are a few things everyone's heard of:

But despite being around for over 100 years now, General Relativity2; (GR) - 2;Einstein's greatest achievement2; - 2;is largely mysterious to everyone from the general public to undergraduate and graduate students in physics.


Could you one day write a story explaining to a lay person what the metric is in GR?

Before we get to "the metric," let's start at the beginning, and talk about how we conceptualize the Universe in the first place.

At a fundamental level, the Universe is made up of quanta2; - 2;entities with physical properties like mass, charge, momentum, etc.2; - 2;that can interact with each other.

  • A quantum can be a particle, a wave, or anything in some weird in-between state, depending on how you look at it.


  • Two or more quanta can bind together, building up complex structures like protons, atoms, molecules or human beings, and all of that is fine.

Quantum physics might be relatively new, having been founded in mostly the 20th century, but the idea that the Universe was made of indivisible entities that interacted with each other goes back more than 2000 years, to at least Democritus of Abdera.

But no matter what the Universe is made of, the things it's composed of need a stage to move on if they're going to interact.


Newton's law of Universal Gravitation

has been superseded by Einstein's general relativity,

but relied on the concept of an

instantaneous action (force) at a distance.

Image credit: Wikimedia commons user Dennis Nilsson.

In Newton's Universe, that stage was flat, empty, absolute space.


Space itself was a fixed entity, sort of like a Cartesian grid:

a 3D structure with an x, y and z axis.

Time always passed at the same rate, and was absolute as well. To any observer, particle, wave or quantum anywhere, they should experience space and time exactly the same as one another.


But by the end of the 19th century, it was clear that Newton's conception was flawed.


Particles that moved close to the speed of light experienced time differently (it dilates) and space differently (it contracts) compared to a particle that was either slow-moving or at rest.


A particle's energy or momentum was suddenly frame-dependent, meaning that space and time weren't absolute quantities; the way you experienced the Universe was dependent on your motion through it.


A "light clock" will appear to run different

for observers moving at different relative speeds,

but this is due to the constancy of the speed of light.

Einstein's law of special relativity governs

 how these time and distance transformations take place.

Image credit: John D. Norton


That was where the notion of Einstein's theory of special relativity came from: some things were invariant, like a particle's rest mass or the speed of light, but others transformed depending on how you moved through space and time.


In 1907, Einstein's former professor, Hermann Minkowski, made a brilliant breakthrough:

he showed that you could conceive of space and time in a single formulation.

In one fell swoop, he had developed the formalism of spacetime.


This provided a stage for particles to move through the Universe (relative to one another) and interact with one another, but it didn't include gravity.


The spacetime he had developed2; - 2;still today known as Minkowski space2; - 2;describes all of special relativity, and also provides the backdrop for the vast majority of the quantum field theory calculations we do.


Quantum field theory calculations

are normally done in flat space,

but general relativity goes beyond that

to include curved space.

QFT calculations are far more complex there.

Image credit: SLAC National Accelerator Laboratory.

If there were no such thing as the gravitational force, Minkowski spacetime would do everything we needed.


Spacetime would be simple, uncurved, and would simply provide a stage for matter to move through and interact. The only way you'd ever accelerate would be through an interaction with another particle.


But in our Universe, we do have the gravitational force, and it was Einstein's principle of equivalence that told us that so long as you can't see what's accelerating you, gravitation treats you the same as any other acceleration.


The identical behavior of

a ball falling to the floor in

an accelerated rocket (left) and on Earth (right)

is a demonstration of Einstein's equivalence principle.

Image credit: Wikimedia Commons user Markus Poessel,

retouched by Pbroks13.

It was this revelation, and the development to link this, mathematically, to the Minkowski-an concept of spacetime, that led to general relativity.


The major difference between special relativity's Minkowski space and the curved space that appears in general relativity is the mathematical formalism known as the Metric Tensor, sometimes called Einstein's Metric Tensor or the Riemann Metric.


Bernhard Riemann was a pure mathematician in the 19th century (and a former student of Gauss, perhaps the greatest mathematician of them all), and he gave a formalism for how any fields, lines, arcs, distances, etc., can exist and be well-defined in an arbitrarily curved space of any number of dimensions.


It took Einstein (and a number of collaborators) nearly a decade to cope with the complexities of the math, but all was said and done, we had general relativity:

a theory that described our three-space-and-one-time dimensional Universe, where gravitation existed.


The warping of spacetime by gravitational masses,

as illustrated to represent General Relativity.

Image credit: LIGO/T. Pyle.

Conceptually, the metric tensor defines how spacetime itself is curved.


Its curvature is dependent on the matter, energy and stresses present within it; the contents of your Universe define its spacetime curvature. By the same token, how your Universe is curved tells you how the matter and energy is going to move through it.


We like to think that an object in motion will continue in motion:

Newton's first law.

We conceptualize that as a straight line, but what curved space tells us is that instead an object in motion continuing in motion follows a geodesic, which is a particularly-curved line that corresponds to unaccelerated motion.


Ironically, it's a geodesic, not necessarily a straight line, that is the shortest distance between two points.


This shows up even on cosmic scales, where the curved spacetime due to the presence of extraordinary masses can curve the background light from behind it, sometimes into multiple images.


An example/illustration

of gravitational lensing, and the bending

of starlight due to mass.

Image credit: NASA / STScI


Physically, there are a number of different pieces that contribute to the Metric Tensor in general relativity.


We think of gravity as due to masses: the locations and magnitudes of different masses determine the gravitational force. In general relativity, this corresponds to the mass density and does contribute, but it's one of only 16 components of the Metric Tensor!


There are also pressure components (such as radiation pressure, vacuum pressure or pressures created by fast-moving particles) that contribute, which are three additional contributors (one for each of the three spatial directions) to the Metric Tensor.


And finally, there are six other components that tell us how volumes change and deform in the presence of masses and tidal forces, along with how the shape of a moving body is distorted by those forces.


This applies to everything from a planet like Earth to a neutron star to a massless wave moving through space:

gravitational radiation.


As masses move

through spacetime relative to one another,

they cause the emission of gravitational waves:

ripples through the fabric of space itself.

These ripples are mathematically encoded

in the Metric Tensor.

Image credit: ESO/L. Calšada.

You might have noticed that 1 + 3 + 6 ≠ 16, but 10, and if you did, good eye!


The Metric Tensor may be a 4 Î 4 entity, but it's symmetric, meaning that there are four "diagonal" components (the density and the pressure components), and six off-diagonal components (the volume/deformation components) that are independent; the other six off-diagonal components are then uniquely determined by symmetry.


The metric tells us the relationship between all the matter/energy in the Universe and the curvature of spacetime itself.


In fact, the unique power of general relativity tells us that if you knew where all the matter/energy in the Universe was and what it was doing at any instant, you could determine the entire evolutionary history of the Universe2; - 2;past, present and future2; - 2;for all eternity.


The four possible fates of the Universe,

with the bottom example fitting the data best:

a Universe with dark energy.

Image credit: E. Siegel.

This is how my sub-field of theoretical physics, cosmology, got its start!


The discovery of the expanding Universe, its emergence from the Big Bang and the dark energy-domination that will lead to a cold, empty fate are all only understandable in the context of general relativity, and that means understanding this key relationship:

between matter/energy and spacetime.

The Universe is a play, unfolding every time a particle interacts with another, and spacetime is the stage on which it all takes place.


The one key counterintuitive thing you've got to keep in mind? The stage isn't a constant backdrop for everyone, but it, too, evolves along with the Universe itself.


Part 2
January 08, 2019






In science fiction, space and time warps are a commonplace.


They are used for rapid journeys around the galaxy, or for travel through time. But today's science fiction, is often tomorrow's science fact. So what are the chances for space and time warps.

All material bodies have a certain extension: length, breadth, height. They are variously placed in relation to each other and constitute parts of one or another system. Space is a form of coordination of coexisting objects and states of matter.


It consists in the fact that objects are extraposed to one another (alongside, beside, beneath, above, within, behind, in front, etc.) and have certain quantitative relationships.


The order of coexistence of these objects and their states forms the structure of space.



Material phenomena are characterized by their duration, the sequence of the stages of their motion, their development. Processes may take place either simultaneously, or precede or succeed one another.


Such, for example, is the interrelation between day and night.


The dimension of time can be measured only with the help of certain standards (in seconds. minutes, hours, days, years, centuries, etc.), that is to say, motions that are accepted as being even. The perception of time also allows us to assess the sequence and duration of events.


Depending on our subjective sensations such as merriment or grief, pleasure or boredom, time seems either short or long. Time is a form of coordination of objects and states of matter in their succession. It consists in the fact that every state is a consecutive link in a process and has certain quantitative relations with other states.


The order of succession of these objects and states forms the structure of time.

Space and time are universal forms of the existence of matter, the coordination of objects. The universality of these forms lies in the fact that they are forms of existence of all the objects and processes that have ever existed or will exist in the infinite universe.


Not only the events of the external world, but also all feelings and thoughts take place in space and time. In the material world everything has extension and duration. Space and time have their peculiarities.


Space has three dimensions:

length, breadth and height, but time has only one2; - 2;from the past through the present to the future.

It is inevitable, unrepeatable and irreversible.

Correct understanding of the essence of space and time is closely connected with the scientific picture of the world. Everything is differentiated, broken down into relatively stable extraposed material formations.


The processes that occur in them and condition their conservation (reproduction) and at the same time their transformation, are also differentiated: they constitute the consecutive change of the states of an object.

Space and time exist objectively. Although we may feel how time in its inexorable passage is carrying us away, we can neither halt nor prolong it. We cannot recover a single moment of existence. The flow of time is beyond our control.


We are as helpless in it as a chip of wood in a river.

Dialectics proceeds from acknowledgement of the unity of motion, space, time and matter, which is expressed in the principle that various forms of the structural organization of matter and the levels of this organization are characterized by their specific motion, space and time.


Thus the spatial organization of a crystal differs from that of a blossoming rose.


The time of historical events occurs, is experienced by their participants and is preserved in the memory of mankind and this kind of time differs from the purely physical time of, say, the motion of the celestial bodies.


However, metaphysical thought separates matter from motion, and both of them, from space and time.


Newton, for example, assumed that space was the empty container of things, that it was incorporeal, absolutely penetrable, never influenced anything and was never affected by any influence.



Universal space was considered to be filled with absolutely motionless ether, and moving bodies were thought to encounter an "ethereal wind" like the wind that resists a running person.


Space was allegedly immutable and motionless, its attributes did not depend on anything, even time; nor did they depend on material bodies or their motion. One could remove all bodies from space and space would still exist and retain its attributes.


Newton held the same views about time.


He believed that time flowed by in the same way throughout the universe and this flow did not depend on anything; time was therefore absolute. Like a river, it flowed on of its own accord, heedless of the existence of material processes.

The idea of absolute space and time corresponded to the physical picture of the world, namely the system of views of matter as a set of atoms separated from each other, possessing immutable volume and inertia (mass), and influencing each other instantaneously either at a distance or through contact.


Revision of the physical picture of the world changed the view of space and time.


The discovery of the electromagnetic field and the realization that field could not be reduced to a state of mechanical environment revealed the flaws in the classical picture of the world. It turned out that matter could not be represented as a set of separate, strictly dissociated elements.


The particles of matter are indeed connected with one another in integral systems by fields whose action is transmitted at a finite speed that is equal for any closed system (the speed of light in a vacuum).

It was held previously that if all matter disappeared from the universe, space and time would remain. The theory of relativity, however, maintains that with the disappearance of matter space and time would also disappear.

To sum up, everything in the world is spatial and temporal.


Space and time are absolute. But since these are forms of matter in motion, they are not indifferent to their content. When it moves, an object does not leave an empty form behind it, space is not an apartment that can be let out to such a tenant as matter, and time cannot be compared to some monster that gnaws at things and leaves its tooth marks on them.


Space and time are conditioned by matter, as a form is conditioned by its content, and every level of the motion of matter possesses its space-time structure.


Thus living cells and organisms, in which geometry becomes more complex and the rhythm of time changes, possess special space-time properties. This is biological time. There is also historical time, whose unit may be the replacement of one generation by another, which corresponds to a century.


Depending on our practical needs, historical time is counted in centuries and millennia. The reference point may be certain cultural-historical events or even legends.

The finite and the infinite.

  • Whose imagination has not been stirred by a mysterious sense of the vastness of the universe?


  • What man has looked up at the dark sky glittering with its myriads of stars and not been awed by the glamour of outer space?


  • Whose heart has not been moved by the majestic splendor of the nocturnal heavens?




In our everyday lives, our dealings with everything around us, we encounter finite objects, processes.


The finite means something that has an end, that is limited in space. In everyday practice we may mean by infinity anything very big or very small, depending on the circumstances.


For example, one billion raised to the power of one hundred is in practice an infinite quantity.


Our experience is too limited for us to be able to define infinity. Scientists like to joke that they begin to understand infinity only when they think of human folly. One may throw a spear from a certain point in space and from the place where it lands one may repeat the throw. And one may go on doing this again and again, never reaching any boundary.


No matter how distant a star may be from us we may still go further than that star. The universe is never "boarded up".


Infinity cannot be traversed to its end. Such infinity would be a "false" infinity. True infinity means constant going beyond the limits of the finite. The universe is not given in any cut-and-dried form, it is constantly reproducing itself; it is a reality that is constantly recreated.


The infinite manifests itself in the finite and through the finite. Through the finite we come to an understanding, a knowledge of the infinite. The finite is a constantly appearing and disappearing moment of an infinite process of change.


Change in general is associated with an object's going beyond its spatial, temporal, quantitative and qualitative limits. The very fact of the interaction of things is constant going beyond the limits of finite, individual existence.


In this constant "going beyond oneself" into outer being, lies the infinite nature of the finite.


An object has innumerable relations with other objects. Thereby it acquires an infinite number of properties. And in this sense infinity implies qualitative diversity, realized in space and time.



We have advanced from the scale of the Earth to the expanses of outer space, to time that has no beginning and no end. This is extensive infinity.


We ourselves appear to be standing midway between the infinite expanses of the universe with its worlds that are known or unknown to us and the equally infinite depths of the world of the smallest particles of matter, which is intensive infinity.


We are the junction, as it were, of roads that lead away into the infinitely large and the infinitely small. We are mere specks of dust in comparison with the stars and at the same time we are giants compared to the tiny microorganisms that swarm in every drop of water.

Thought has penetrated from regions describable only in terms of millions of light years to regions that may be measured in trillionths of a centimeter! And there, too, we find the properties of the finite and the infinite.


Thus, many physicists assume the existence of a certain basic length2; - 2;the spatial quantum. It would, they say, be as pointless to consider any smaller length as it would be to consider, for example, a quantity of gold less than one atom, because such a quantity would not even constitute the given chemical element.


So scientists assume the existence of "atoms" of space.


From this follows the recognition of minimal time, beyond whose limits the concept of phase, that is to say, changes of state in time, loses all meaning.

At attempt to refute the theory of the infinity of the universe is to be found in the concept of the "expanding" universe. James Jeans, for example, assumed that not only was the quantity of matter in the universe diminishing, but also that any matter that remained was constantly receding into space at colossal and ominously increasing speed.


And yet there are no valid grounds for such conclusions.


The metagalaxy in which we observe this centrifugal movement of the galaxies, despite its enormous size as it appears to us, is only a tiny particle in the infinite universe, so it cannot be assumed that the whole universe is "expanding".



To sum up, all objects and processes in the world are finite. But the totality of finite things and processes is infinite. The universe had no beginning, has no end and is inexhaustible.


Beyond the most distant stellar systems that modern science and technology have permitted us to observe there are still other gigantic celestial bodies. And so on ad infinitum. There are no limits beyond which there might be something that cannot be embraced by the concept of objective reality and there is nothing above it or outside it.


Objective reality is in everything. It is everything...


The concept of limit has meaning only when applied to the finite. Neither our distance-bound imagination nor the spacemen of the future can ever encounter some supernatural obstacle such as non-existence. They will never run into something that differs from matter.


No matter how much time passes prior to some event, time will go on after it. No matter how long ago a certain event took place, it was preceded by countless other events. The chain of events has never been broken. Its links are numberless.


In the universe as a whole there is no initial or culminating point; the universe is equally open at both ends. If time were finite, the world must have had a beginning.


To acknowledge the beginning of the world's existence in time would be to acknowledge creation and, consequently, a creator.

The concept of beginning is meaningful when applied not to the universe as a whole but only to separate, specific things and processes, that is to say, to the finite. We can set no limits to the universe as a whole. It categorically forbids us to do so. It is ageless. It is infinitely old and eternally young.


Someone once wittily remarked that he could not imagine the universe having lived its life and sadly vegetating for the rest of eternity.

It seems that what happens, is that when space-time gets warped almost enough to allow travel into the past, virtual particles can almost become real particles, following closed trajectories. The density of the virtual particles, and their energy, become very large. This means that the probability of these histories is very low.


Thus it seems there may be a Chronology Protection Agency at work, making the world safe for historians.


But this subject of space and time warps is still in its infancy. According to string theory, which is our best hope of uniting General Relativity and Quantum Theory, into a Theory of Everything, space-time ought to have ten dimensions, not just the four that we experience.


The idea is that six of these ten dimensions are curled up into a space so small, that we don't notice them.


On the other hand, the remaining four directions are fairly flat, and are what we call space-time. If this picture is correct, it might be possible to arrange that the four flat directions got mixed up with the six highly curved or warped directions.


What this would give rise to, we don't yet know. But it opens exciting possibilities.

The conclusion of this lecture is that rapid space-travel, or travel back in time, can't be ruled out, according to our present understanding.


They would cause great logical problems, so let's hope there's a Chronology Protection Law, to prevent people going back, and killing our parents.